Context. To understand the formation of planetary systems, it is important to understand the initial conditions of planet formation, that is, the young gas-rich planet forming disks. Spatially resolved, high-contrast observations are of particular interest since substructures in disks that are linked to planet formation can be detected. In addition, we have the opportunity to reveal close companions or even planets in formation that are embedded in the disk. Aims: In this study, we present the first results of the Disk Evolution Study Through Imaging of Nearby Young Stars (DESTINYS), an ESO/SPHERE large program that is aimed at studying disk evolution in scattered light, mainly focusing on a sample of low-mass stars (< 1 M⊙) in nearby (∼...